Numerical and Analytical Study of Concrete Beams Reinforced with Hybrid Fiber-Reinforced Polymer and Steel Bars

Acknowledgments

Notation

A_cf,eff

effective concrete area of FRP bars (mm²);

A_cs,eff

effective concrete area of steel bars (mm²);

A_f

area of tensile GFRP bars (mm²);

A_s

area of tensile steel bars (mm²);

a

shear span of the beam (mm);

a_f

distance from the bottom surface of the beam to the center of tensile FRP bars (mm);

a_s

distance from the bottom surface of the beam to the center of tensile steel bars (mm);

$a_s^{\mathrm{\prime }}$

distance from the top surface of the beam to the center of compressive bars (mm);

b

width of the beam (mm);

c

neutral axis depth (mm);

d

distance from the extreme compression fiber to the centroid of the tensile reinforcement (mm);

E_c

modulus of elasticity of concrete (MPa);

E_f

modulus of elasticity of GFRP bars (MPa);

E_s

modulus of elasticity of steel bars (MPa);

E_t

tensile modulus of elasticity of concrete (MPa), assumed to be the same as E_c;

$f_c^{\mathrm{\prime }}$

cylinder compressive strength of concrete (MPa);

f_f

stress of FRP bars when concrete crushing failure occurs (MPa);

f_fu

tensile strength of FRP bars (MPa);

f_tu

tensile strength of concrete (MPa);

f_u

ultimate strength of steel bars (MPa);

f_y

yield strength of steel bars (MPa);

h

total depth of the beam (mm);

I_cr

moment of inertia of transformed cracked section (mm⁴);

I_cr(1)

moment of inertia of transformed cracked section before steel yielding (mm⁴);

I_cr(2)

moment of inertia of transformed cracked section after steel yielding (mm⁴);

I_e

effective moment of inertia (mm⁴);

I_e(2)

effective moment of inertia after steel yielding (mm⁴);

I_g

moment of inertia of the gross concrete section about the centroidal axis, neglecting reinforcement (mm⁴);

L

clear span of the beam (mm);

M_a

moment due to the applied load (kN · m);

M_cr

cracking moment (kN · m);

M_cr,e

experimental cracking moment (kN · m);

M_u

ultimate moment capacity (kN · m);

M_u,e

experimental ultimate moment capacity (kN · m);

M_y

yielding moment (kN · m);

M_y,e

experimental yielding moment (kN · m);

M_0.001

moment at a service limit state with the corresponding concrete strain ɛ_c = 0.001;

P

applied load (kN);

S_w

average crack spacing at constant moment region (mm);

s

slip (relative displacement between steel and concrete cross-sections);

s_f,i

slip of FRP bars at node i;

s_s,i

slip of steel bars at node i;

s₁

slip corresponding to the peak bond stress;

U_f

perimeter of FRP bars (mm);

U_s

perimeter of steel bars (mm);

y_t

distance from centroidal axis of gross section, neglecting reinforcement, to tension face (mm);

γ

factor determined by load and boundary conditions;

δ_i

deflection at node i;

ɛ_c

compressive strain in concrete;

ɛ_ci

compressive strain in concrete at the ith layer;

ɛ_co

strain in concrete at maximum compressive stress;

ɛ_ct

concrete strain at the top layer;

ɛ_cu

ultimate strain in concrete;

ɛ_f

strain in tensile FRP bars;

ɛ_fu

design rupture strain of FRP reinforcement;

ɛ_s

strain in tensile steel bars;

ɛ_sh

strain at strain hardening for steel bars;

${\epsilon }_s^{\mathrm{\prime }}$

strain in compressive steel bars;

ɛ_t

tensile strain in concrete;

ɛ_ti

tensile strain in concrete at the ith layer;

ɛ_tu

tensile strain in concrete at tensile stress of f_tu;

ɛ_u

strain at ultimate stress for steel bars;

ɛ_y

yield strain in steel bars;

θ_i

rotation at node i;

λ

modification factor which accounts for the reduced mechanical properties of lightweight concrete compared with normalweight concrete of the same compressive strength;

μ

coefficient determining the decay rate of tensile strength;

μ_E,Δ

deformability index;

ρ_eff

effective reinforcement ratio;

ρ_f

ratio of A_f to bd;

ρ_s

ratio of A_s to bd;

σ_c

compressive stress of concrete (MPa);

σ_cf,i

stress of concrete between the (i − 1)th and ith nodes of FRP bars (MPa);

σ_cfm,i,i−1

mean stress of concrete between the (i − 1)th and ith nodes of FRP bars (MPa);

σ_csm,i,i−1

mean stress of concrete between the (i − 1)th and ith nodes of steel bars (MPa);

σ_f,i

stress of FRP bars at node i (MPa);

σ_fm,i,i−1

mean stress of FRP bars between the (i − 1)th and ith nodes (MPa);

σ_s,i

stress of steel bars at node i (MPa);

σ_sm,i,i−1

mean stress of steel bars between the (i − 1)th and ith nodes (MPa);

τ

bond stress (MPa);

τ_f,i

bond stress of FRP bars at node i (MPa);

τ_max

peak bond stress for reinforcing bars (MPa);

τ_s,i

bond stress of steel bars at node i (MPa);

φ_i

curvature at node i;

φ_u

curvature at the ultimate limit state; and

φ_0.001

curvature at a service limit state with the corresponding concrete strain ɛ_c = 0.001.

References